Natural vibrations of anisotropic plates with several internal line hinges

2012

The present paper deals with the free transverse vibration of a tapered anisotropic plate with several arbitrarily located internal line hinges and non-smooth boundary, elastically restrained against rotation and translation. The equations of motion and its associated boundary and transition conditions are rigorously derived using Hamilton’s principle. The governing eigenvalue problem is solved employing a combination of the Ritz method and the Lagrange multipliers method. The deflections of the plate and the Lagrange multipliers are approximated by polynomials as coordinate functions. The developed algorithm allows obtaining approximate solutions for plates with different geometries and boundary conditions, including edges and line hinges elastically restrained. In order to obtain an indication of the accuracy of the developed mathematical model, some cases available in the literature are considered. New results are presented for different boundary conditions and restraint conditio...

Applied Acoustics, 2012

This paper deals with the study of free transverse vibrations of rectangular plates with an internal line hinge and elastically restrained boundaries. The equations of motion and its associated boundary and transition conditions are rigorously derived using Hamilton's principle. The governing eigenvalue equation is solved employing a combination of the Ritz method and the Lagrange multipliers method. The deflections of the plate and the Lagrange multipliers are approximated by polynomials as coordinate functions. The developed algorithm allows obtaining approximate solutions for plates with different aspect ratios, boundary conditions, including edges elastically restrained by both translational and rotational springs, and arbitrary locations of the line hinge. Therefore, a unified algorithm has been implemented. Sets of parametric studies are performed and the results are given in graphical and tabular form.

Acta Mechanica, 2011

This paper deals with the formulation of an analytical model for the dynamic behavior of anisotropic plates, with an arbitrarily located internal line hinge with elastic supports and piecewise smooth boundaries elastically restrained against rotation and translation among other complicating effects. The equations of motion and its associated boundary and transition conditions are derived using Hamilton's principle. By introducing an adequate change of variables, the energies that correspond to the different elastic restraints are handled in a general framework. The concept of transition conditions and the determination of the analytical expressions are presented. Analytical examples are worked out to illustrate the range of applications of the developed analytical model. One of the essential features of this work is to demonstrate how the commonly formal derivations used in the applications of the calculus of variations can be made rigorous.

Computers & Structures, 1978

Journal of Sound and Vibration, 1997

The free vibrations of a circular plate having elastic constraints variable according to the angular coordinate are investigated. The non-uniform translational and rotational stiffness of the constraints are expanded in a Fourier series; it is assumed that the system presents a symmetry axis. The mode shapes are expanded in a Fourier-Bessel series by using the Rayleigh-Ritz method. The eigenfunctions of the free-edge circular plate vibrating in vacuum are assumed as admissible functions. This choice allows one to compute the potential energy of the plate using the kinetic energy of single modes of free-edge plates. The effect of the in-plane load is included and internal constraints are studied. By using the same technique, the free vibrations of a circular plate resting on an annular, non-uniform, Winkler foundation are investigated. Numerical results are given for the cases studied already, in order to validate the proposed method, and for bolted (or riveted) plates fixed by different numbers of bolts.

International Journal of Non-linear Mechanics, 2006

This paper is concerned with a new improved formulation of the theoretical model previously developed by Benamar et al. based on Hamilton's principle and spectral analysis, for the geometrically non-linear vibrations of thin structures. The problem is reduced to a non-linear algebraic system, the solution of which leads to determination of the amplitude-dependent fundamental non-linear mode shapes, the frequency parameters, and the non-linear stress distributions. The cases of C-S-C-S and C-S-S-S rectangular plates are examined, and the results obtained are in a good qualitative and quantitative agreement with the previous available works, based on various methods. In order to obtain explicit analytical solutions for the first non-linear mode shapes of C-S-C-S RP 2 and C-S-S-S RP, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. For beams and fully clamped rectangular plates, has been slightly modified, and adapted to the above cases, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values up to 0.75 and 0.6 for the first non-linear mode shapes of C-S-C-S RP and C-S-S-S RP, respectively.

Proceedings of the Institution of Mechanical …, 1994

This paper considers linear free vibrations of thin isotropic rectangular plates with combinations of the classical boundary conditions of simply supported, clamped and free edges and the mathematically possible condition of guided edges. The total number of ...

Journal of Sound and Vibration, 2000

Mechanics Research Communications, 2010

A Ritz method using a set of trigonometric functions is developed to obtain accurate in-plane modal properties of rectangular plates with arbitrary nonuniform elastic edge restraints. Reliability of the current approach is first assured by comparison against exact and analytical-type solutions for plates with classical boundary conditions and uniform elastic boundaries. For the first time to the author's knowledge, the problem of free in-plane vibration of plates having triangularly and parabolically varying elastic edge supports is then considered. Accurate upper-bound solutions are tabulated to provide valuable benchmark data against which the findings of other researchers can be compared in the future. Effects of nonuniform elastic spring stiffness on the in-plane natural frequencies and modal shapes are also presented.

Free vibration of transversely isotropic circular plates with various edge boundary conditions is analyzed on the basis of theory of elasticity without a priori plate assumptions. The governing equations for vibration of transversely isotropic circular plates are derived from the three-dimensional equations of elasticity in the cylindrical coordinates. By means of separation of variables, two sets of solutions are obtained, which enable us to satisfy various edge boundary conditions of the problems and determine the natural frequencies of the circular plates. Numerical results for three kinds of edge conditions are evaluated and compared with those obtained according to 2D plate theories such as DQM, HSDT and Mindlin's solutions. The study shows that our results can find much more natural frequencies of circular plates than the 2D solutions for circular plates. These 2D solutions are applicable while the ratios of thickness to radius of the plates are lesser than 0.05 and the edge conditions are clamped or simply-supported.